If a compression-type faucet continues to drip after both handles are turned to the off position, remove both handles to inspect for damage to the stem assemblies.
Tip: When repairing a faucet, close the sink's stopper and cover it with a rag to prevent small parts from falling into the drain.
Turn the water shut-off valves under the sink to the off position, then turn the faucet handles to the on position to drain any remaining water.
If there is a decorative cap on top of the handle, carefully pop it off with a utility knife (Image 1) to expose the screw holding the handle in place. Remove the screw with a Phillips-head screwdriver, and pull off the handle to expose the stem assembly. (You may need to pry the handle off gently with a screwdriver. If so, pad the screwdriver with a rag to avoid damaging the faucet.) With the handle out of the way, use slip-joint pliers to remove the retaining nut that holds the stem in place (Image 2).
Inspect the stem assembly for damage. Check the rubber washer on the end. If it looks OK, unscrew the packing nut to see whether there is damage to the O-rings inside (Image 1). If the washer is damaged, remove the screw, and replace the old washer with a new one. If an O-ring is bad, cut it off with a utility knife, coat a new one with heat-proof grease, and slide the new O-ring into place on the stem, using a flat-head screwdriver as a guide (Image 2).
An alternative to replacing washers and rings is to replace the entire stem assembly. If you decide to take this course, it's best to replace the stems on both the hot- and cold-water sides. Note that because the hot- and cold-water handles often turn in opposite directions, the hot and cold stem assemblies may be different. Make sure to buy the right parts. Some are color-coded to indicate hot or cold.
If the washer and O-rings look good, use your finger to check for rough spots around the valve seat inside the faucet (Image 1). If the seat is damaged, the faucet may not make a watertight seal even with new washers. Use a specially made seat wrench to remove damaged seats and replace them (Image 2).
Once repairs are complete, reassemble the faucet: drop in the repaired (or new) stem assembly and tighten the retaining nut with slip-joint pliers. Replace the handles and secure them with their screws. Replace the decorative cap.
Turn the water supply back on, and check the hot- and cold-water sides to make sure they both work properly and without drips.